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Transmission Lines

Transmission Lines. Transmission line effects must be considered when length is comparable to ¼ wavelength We will ignore the energy loss on transmission lines Concentrate on time-domain description rather than frequency domain. i. z g. +. z l. e g. e. -. i. x. d. l.

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Transmission Lines

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  1. Transmission Lines • Transmission line effects must be considered when length is comparable to ¼ wavelength • We will ignore the energy loss on transmission lines • Concentrate on time-domain description rather than frequency domain i zg + zl eg e - i x d l ECE 766 Computer Interfacing and Protocols

  2. C: Capacitance / Unit Length [F/m] L: Inductance / Unit Length [H/m] R: Conductor Resistance / Unit Length [Ω/m] G: Insulation Conductance /Unit Length [ /m] Note G ≠ 1/R ! Ω Transmission Lines + + - - ECE 766 Computer Interfacing and Protocols

  3. By KVL: In the limit as : Transmission Lines ECE 766 Computer Interfacing and Protocols

  4. Transmission Lines By KCL: In the limit as : ECE 766 Computer Interfacing and Protocols

  5. Lossless Case Take partial derivative w.r.t. x in (1) and partial derivative w.r.t. t in (2),then substitute. Recognize as wave equations ECE 766 Computer Interfacing and Protocols

  6. Show that the solution is in the form . Lossless Case ECE 766 Computer Interfacing and Protocols

  7. Interpretation • is a wave traveling to right with velocity • is a wave traveling to left with velocity • Solving for i, we obtain • is characteristic of line ECE 766 Computer Interfacing and Protocols

  8. Interpretation Note: Each traveling wave direction, e and i are related by z0. ECE 766 Computer Interfacing and Protocols

  9. il + + z0 el zl e - - Reflections • Look at terminations with real impedances frequency independent • Wave of voltage and current traveling to right • At termination ECE 766 Computer Interfacing and Protocols

  10. Reflections • Hence, there must be reflected waves e- and i- such that • In terms of voltage Reflection coefficient ECE 766 Computer Interfacing and Protocols

  11. Reflections • Special cases: • zl = z0 k=0Matched, no reflections. Line looks infinite. • zl = 0, short circuit k = -z0/z0 = -1 • zl = , open circuit k = 1 8 ECE 766 Computer Interfacing and Protocols

  12. t=0 R=3z0 z0 E x=0 x=l t=.3T E 3E/2 t=1.3T E 3E/2 t=2.3T E 1.5E 1.125E Voltage at load E .9375E .75E 7T T 2T 3T 4T 5T 6T Multiple Reflections Hitting load, a wave of E/2 is produced Arriving wave of E/2 isreflected toward load ECE 766 Computer Interfacing and Protocols

  13. x k0=-1 kl=.5 e=0 E e=E T E/2 e=3E/2 2T Time -E/2 e=E 3T -E/4 e=3E/4 4T E/4 e=E 5T e=9E/8 E/8 Time-Space (Bounce) Diagram • Mark reflection coefficients • Write initial voltages • Write wave amplitudes • Update • Wave amplitudes when reflected • Voltages as waves cross ECE 766 Computer Interfacing and Protocols

  14. x k0=.5 kl=1 t=0 R=3z0 e=0 E/4 z0 E e=E/4 T l k0=.5 kl=1 E/4 e=E/2 2T 3z0 Time E/8 e=5E/8 z0 E 3T E/8 e=3E/4 E 4T At x=0 E/16 e=13E/16 5T e=7E/8 E/16 t 4T T 2T 3T 5T Example Magnitude of the first wave: ECE 766 Computer Interfacing and Protocols

  15. Reflections in Digital Lines • Consider one source and one load • Option 1:Do not terminate either end. Ringing will stop eventually. • Pro: Simple, no additional power loss • Con: Limited speed ECE 766 Computer Interfacing and Protocols

  16. z0 Reflections in Digital Lines • Option 2:Matched termination at the end • Pro: No reflections • Con: Excessive power consumptionFor z0=150Ω, power consumption 135mWReduce by duty factor (0.5 for regular lines, 0.05 for floppy drives)Multiply by number of lines Open-collectordriver ECE 766 Computer Interfacing and Protocols

  17. z0 Reflections in Digital Lines • Option 3:Matched termination at the source end • Pro: Can be run at the same speed as load termination If receiver has very high input impedance, full voltage appears at the receiver No power dissipated at constant voltage level • Con: Special, high input impedance line receivers required (not suited for standard TTL) Look at multiple terminations ECE 766 Computer Interfacing and Protocols

  18. x k0=0 kl=1 e=0 z0 E/2 e=.5E T E E/2 e=E 2T E At x=.5L t 2T .5T T 1.5T Reflection in Digital Lines • Problem for multiple receivers • Assume the ideal case, where taps are infinitely short and have infinite impedances • Even for this case, intermediate taps do not get full signal immediately ECE 766 Computer Interfacing and Protocols

  19. e2+, i2+ e1+, i1+ e1-, i1- e3+, i3+ Reflection and Transmission at Junctions ECE 766 Computer Interfacing and Protocols

  20. k=1 3/2 L z0 t=0 k=0 L k=1 2/3 L E/2 T -E/6 E/3 E/3 2T E/3 2E/9 2E/9 3T -E/9 E/3 4T Reflection and Transmission at Junctions Many multiple reflections,eventually come to rest ECE 766 Computer Interfacing and Protocols

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